The geometric classification of nilpotent commutative $\mathfrak{C}\mathfrak{D}$-algebras

We give a geometric classification of complex $5$-dimensional nilpotent commutative $\mathfrak{C}\mathfrak{D}$-algebras. The corresponding geometric variety has dimension $24$ and decomposes into $10$ irreducible components determined by the Zariski closures of a two-parameter family of algebras, three one-parameter families of algebras, and $6$ rigid algebras.